# Operator.m

Generates Hilbert space operators and Liouville space superoperators from their human-readable descriptions. Syntax:

A=operator(spin_system,operators,spins,operator_type)

The function supports two types of calls:

1. If an operator is given as a single string and spins are named by passing a single string, e.g.

sum_Lz=operator(spin_system,'Lz','13C');

then the function returns the sum of the corresponding single-spin operators (in Hilbert space) or superoperators (in Liouville space) on all spins with that name. Valid labels for operators in this type of call are

'E', 'Lz', 'L+', 'L-', 'Tl,m'

where l and m are integers. In the latter case a spherical tensor operator or superoperator is returned. Valid labels for spins are standard isotope names as well as

'electrons', 'nuclei', 'all'

2. If the operators are supplied as a cell array of strings and spins as a cell array of numbers, a product operator (in Hilbert space) or superoperator (in Liouville space) is produced, e.g.

LzSp=operator(spin_system,{'Lz','L+'},{1,2});

will return LzL+ operator in Hilbert space and [LzL+,_] commutation superoperator in Liouville space, with Lz on spin 1 and L+ on spin 2. Valid labels for operators in the cell array are

'E', 'Lz', 'L+', 'L-', 'Tl,m'

where l and m are integers. In the latter case a spherical tensor operator or superoperator is included into the product.

In Liouville space calculations, operator_type can be set to:

'left' - produces left side product superoperator

'right' - produces right side product superoperator

'comm' - produces commutation superoperator (default)

'acomm' - produces anticommutation superoperator

In Hilbert space calculations operator_type parameter is not permitted.

**WARNING**: do not try to obtain product commutation superoperators by multiplying them up! It is easy to see that

[math]{{\hat{\hat{O}}}^{2}}=\left[ \hat{O},\left[ \hat{O},\_ \right] \right]\ne \left[ {{{\hat{O}}}^{2}},\_ \right][/math]

If you require a commutation superoperator corresponding to a multi-spin operator, use the syntax given in the Section 2 above.

## Examples

**1. L+ on spin 3**

Lp=operator(spin_system,{'L+'},{3});

An operator will be generated in Hilbert space and a commutation superoperator in Liouville space.

**2. A sum of Lx on all 15N spins in the system**

Lx=(operator(spin_system,'L+','15N')+operator(spin_system,'L-','15N'))/2;

An operator will be generated in Hilbert space and a commutation superoperator in Liouville space.

**3. AxBx between spin 2 and spin 5**

Both components are Cartesian and must therefore be translated into the convention above:

The Spinach code would therefore be:

AxBx=(operator(spin_system,{'L+','L+'},{2,5})+... operator(spin_system,{'L+','L-'},{2,5})+... operator(spin_system,{'L-','L+'},{2,5})+... operator(spin_system,{'L-','L-'},{2,5}))/4;

An operator will be generated in Hilbert space and a commutation superoperator in Liouville space.

## See also

unit_state.m, unit_oper.m, mprealloc.m, singlet.m, equilibrium.m, state.m

*Version 1.9, authors: Ilya Kuprov, Luke Edwards, Dmitry Savostyanov*